def Kautz(self, k, D, vertices='strings'):
r"""
Returns the Kautz digraph of degree `d` and diameter `D`.
The Kautz digraph has been defined in [Kautz68]_. The Kautz digraph of
degree `d` and diameter `D` has `d^{D-1}(d+1)` vertices. This digraph is
build upon a set of vertices equal to the set of words of length `D`
from an alphabet of `d+1` letters such that consecutive letters are
differents. There is an arc from vertex `u` to vertex `v` if `v` can be
obtained from `u` by removing the leftmost letter and adding a new
letter, distinct from the rightmost letter of `u`, at the right end.
The Kautz digraph of degree `d` and diameter `D` is isomorphic to the
digraph of Imase and Itoh [II83]_ of degree `d` and order
`d^{D-1}(d+1)`.
See also the
:wikipedia:`Wikipedia article on Kautz Graphs <Kautz_graph>`.
INPUTS:
- ``k`` -- Two possibilities for this parameter :
- An integer equal to the degree of the digraph to be produced, that
is the cardinality minus one of the alphabet to use.
- An iterable object to be used as the set of letters. The degree of
the resulting digraph is the cardinality of the set of letters
minus one.
- ``D`` -- An integer equal to the diameter of the digraph, and also to
the length of a vertex label when ``vertices == 'strings'``.
- ``vertices`` -- 'strings' (default) or 'integers', specifying whether
the vertices are words build upon an alphabet or integers.
EXAMPLES::
sage: K = digraphs.Kautz(2, 3)
sage: K.is_isomorphic(digraphs.ImaseItoh(12, 2), certify = True)
(True,
{'010': 0,
'012': 1,
'020': 3,
'021': 2,
'101': 11,
'102': 10,
'120': 9,
'121': 8,
'201': 5,
'202': 4,
'210': 6,
'212': 7})
sage: K = digraphs.Kautz([1,'a','B'], 2)
sage: K.edges()
[('1B', 'B1', '1'), ('1B', 'Ba', 'a'), ('1a', 'a1', '1'), ('1a', 'aB', 'B'), ('B1', '1B', 'B'), ('B1', '1a', 'a'), ('Ba', 'a1', '1'), ('Ba', 'aB', 'B'), ('a1', '1B', 'B'), ('a1', '1a', 'a'), ('aB', 'B1', '1'), ('aB', 'Ba', 'a')]
sage: K = digraphs.Kautz([1,'aA','BB'], 2)
sage: K.edges()
[('1,BB', 'BB,1', '1'), ('1,BB', 'BB,aA', 'aA'), ('1,aA', 'aA,1', '1'), ('1,aA', 'aA,BB', 'BB'), ('BB,1', '1,BB', 'BB'), ('BB,1', '1,aA', 'aA'), ('BB,aA', 'aA,1', '1'), ('BB,aA', 'aA,BB', 'BB'), ('aA,1', '1,BB', 'BB'), ('aA,1', '1,aA', 'aA'), ('aA,BB', 'BB,1', '1'), ('aA,BB', 'BB,aA', 'aA')]
TESTS:
An exception is raised when the degree is less than one::
sage: G = digraphs.Kautz(0, 2)
Traceback (most recent call last):
...
ValueError: Kautz digraphs are defined for degree at least one.
sage: G = digraphs.Kautz(['a'], 2)
Traceback (most recent call last):
...
ValueError: Kautz digraphs are defined for degree at least one.
An exception is raised when the diameter of the graph is less than one::
sage: G = digraphs.Kautz(2, 0)
Traceback (most recent call last):
...
ValueError: Kautz digraphs are defined for diameter at least one.
REFERENCE:
.. [Kautz68] W. H. Kautz. Bounds on directed (d, k) graphs. Theory of
cellular logic networks and machines, AFCRL-68-0668, SRI Project 7258,
Final Rep., pp. 20-28, 1968.
"""
if D < 1:
raise ValueError(
"Kautz digraphs are defined for diameter at least one.")
from sage.combinat.words.words import Words
from sage.rings.integer import Integer
my_alphabet = Words(
[str(i) for i in range(k + 1)] if isinstance(k, Integer) else k, 1)
if my_alphabet.size_of_alphabet() < 2:
raise ValueError(
"Kautz digraphs are defined for degree at least one.")
if vertices == 'strings':
# We start building the set of vertices
V = [i for i in my_alphabet]
for i in range(D - 1):
VV = []
for w in V:
VV += [w * a for a in my_alphabet if not w.has_suffix(a)]
V = VV
# We now build the set of arcs
G = DiGraph()
for u in V:
for a in my_alphabet:
if not u.has_suffix(a):
G.add_edge(u.string_rep(), (u[1:] * a).string_rep(),
a.string_rep())
else:
d = my_alphabet.size_of_alphabet() - 1
G = digraphs.ImaseItoh((d + 1) * (d**(D - 1)), d)
G.name("Kautz digraph (k=%s, D=%s)" % (k, D))
return G
def Kautz(self, k, D, vertices = 'strings'):
r"""
Returns the Kautz digraph of degree `d` and diameter `D`.
The Kautz digraph has been defined in [Kautz68]_. The Kautz digraph of
degree `d` and diameter `D` has `d^{D-1}(d+1)` vertices. This digraph is
build upon a set of vertices equal to the set of words of length `D`
from an alphabet of `d+1` letters such that consecutive letters are
differents. There is an arc from vertex `u` to vertex `v` if `v` can be
obtained from `u` by removing the leftmost letter and adding a new
letter, distinct from the rightmost letter of `u`, at the right end.
The Kautz digraph of degree `d` and diameter `D` is isomorphic to the
digraph of Imase and Itoh [II83]_ of degree `d` and order
`d^{D-1}(d+1)`.
See also the
:wikipedia:`Wikipedia article on Kautz Graphs <Kautz_graph>`.
INPUTS:
- ``k`` -- Two possibilities for this parameter :
- An integer equal to the degree of the digraph to be produced, that
is the cardinality minus one of the alphabet to use.
- An iterable object to be used as the set of letters. The degree of
the resulting digraph is the cardinality of the set of letters
minus one.
- ``D`` -- An integer equal to the diameter of the digraph, and also to
the length of a vertex label when ``vertices == 'strings'``.
- ``vertices`` -- 'strings' (default) or 'integers', specifying whether
the vertices are words build upon an alphabet or integers.
EXAMPLES::
sage: K = digraphs.Kautz(2, 3)
sage: K.is_isomorphic(digraphs.ImaseItoh(12, 2), certify = True)
(True,
{'010': 0,
'012': 1,
'020': 3,
'021': 2,
'101': 11,
'102': 10,
'120': 9,
'121': 8,
'201': 5,
'202': 4,
'210': 6,
'212': 7})
sage: K = digraphs.Kautz([1,'a','B'], 2)
sage: K.edges()
[('1B', 'B1', '1'), ('1B', 'Ba', 'a'), ('1a', 'a1', '1'), ('1a', 'aB', 'B'), ('B1', '1B', 'B'), ('B1', '1a', 'a'), ('Ba', 'a1', '1'), ('Ba', 'aB', 'B'), ('a1', '1B', 'B'), ('a1', '1a', 'a'), ('aB', 'B1', '1'), ('aB', 'Ba', 'a')]
sage: K = digraphs.Kautz([1,'aA','BB'], 2)
sage: K.edges()
[('1,BB', 'BB,1', '1'), ('1,BB', 'BB,aA', 'aA'), ('1,aA', 'aA,1', '1'), ('1,aA', 'aA,BB', 'BB'), ('BB,1', '1,BB', 'BB'), ('BB,1', '1,aA', 'aA'), ('BB,aA', 'aA,1', '1'), ('BB,aA', 'aA,BB', 'BB'), ('aA,1', '1,BB', 'BB'), ('aA,1', '1,aA', 'aA'), ('aA,BB', 'BB,1', '1'), ('aA,BB', 'BB,aA', 'aA')]
TESTS:
An exception is raised when the degree is less than one::
sage: G = digraphs.Kautz(0, 2)
Traceback (most recent call last):
...
ValueError: Kautz digraphs are defined for degree at least one.
sage: G = digraphs.Kautz(['a'], 2)
Traceback (most recent call last):
...
ValueError: Kautz digraphs are defined for degree at least one.
An exception is raised when the diameter of the graph is less than one::
sage: G = digraphs.Kautz(2, 0)
Traceback (most recent call last):
...
ValueError: Kautz digraphs are defined for diameter at least one.
REFERENCE:
.. [Kautz68] W. H. Kautz. Bounds on directed (d, k) graphs. Theory of
cellular logic networks and machines, AFCRL-68-0668, SRI Project 7258,
Final Rep., pp. 20-28, 1968.
"""
if D < 1:
raise ValueError("Kautz digraphs are defined for diameter at least one.")
from sage.combinat.words.words import Words
from sage.rings.integer import Integer
my_alphabet = Words([str(i) for i in range(k+1)] if isinstance(k, Integer) else k, 1)
if my_alphabet.size_of_alphabet() < 2:
raise ValueError("Kautz digraphs are defined for degree at least one.")
if vertices == 'strings':
# We start building the set of vertices
V = [i for i in my_alphabet]
for i in range(D-1):
VV = []
for w in V:
VV += [w*a for a in my_alphabet if not w.has_suffix(a) ]
V = VV
# We now build the set of arcs
G = DiGraph()
for u in V:
for a in my_alphabet:
if not u.has_suffix(a):
G.add_edge(u.string_rep(), (u[1:]*a).string_rep(), a.string_rep())
else:
d = my_alphabet.size_of_alphabet()-1
G = digraphs.ImaseItoh( (d+1)*(d**(D-1)), d)
G.name( "Kautz digraph (k=%s, D=%s)"%(k,D) )
return G
def DeBruijn(self, k, n, vertices='strings'):
r"""
Returns the De Bruijn digraph with parameters `k,n`.
The De Bruijn digraph with parameters `k,n` is built upon a set of
vertices equal to the set of words of length `n` from a dictionary of
`k` letters.
In this digraph, there is an arc `w_1w_2` if `w_2` can be obtained from
`w_1` by removing the leftmost letter and adding a new letter at its
right end. For more information, see the
:wikipedia:`Wikipedia article on De Bruijn graph <De_Bruijn_graph>`.
INPUT:
- ``k`` -- Two possibilities for this parameter :
- An integer equal to the cardinality of the alphabet to use, that
is the degree of the digraph to be produced.
- An iterable object to be used as the set of letters. The degree
of the resulting digraph is the cardinality of the set of
letters.
- ``n`` -- An integer equal to the length of words in the De Bruijn
digraph when ``vertices == 'strings'``, and also to the diameter of
the digraph.
- ``vertices`` -- 'strings' (default) or 'integers', specifying whether
the vertices are words build upon an alphabet or integers.
EXAMPLES::
sage: db=digraphs.DeBruijn(2,2); db
De Bruijn digraph (k=2, n=2): Looped digraph on 4 vertices
sage: db.order()
4
sage: db.size()
8
TESTS::
sage: digraphs.DeBruijn(5,0)
De Bruijn digraph (k=5, n=0): Looped multi-digraph on 1 vertex
sage: digraphs.DeBruijn(0,0)
De Bruijn digraph (k=0, n=0): Looped multi-digraph on 0 vertices
"""
from sage.combinat.words.words import Words
from sage.rings.integer import Integer
W = Words(range(k) if isinstance(k, Integer) else k, n)
A = Words(range(k) if isinstance(k, Integer) else k, 1)
g = DiGraph(loops=True)
if vertices == 'strings':
if n == 0:
g.allow_multiple_edges(True)
v = W[0]
for a in A:
g.add_edge(v.string_rep(), v.string_rep(), a.string_rep())
else:
for w in W:
ww = w[1:]
for a in A:
g.add_edge(w.string_rep(), (ww * a).string_rep(),
a.string_rep())
else:
d = W.size_of_alphabet()
g = digraphs.GeneralizedDeBruijn(d**n, d)
g.name("De Bruijn digraph (k=%s, n=%s)" % (k, n))
return g
def DeBruijn(self, k, n, vertices = 'strings'):
r"""
Returns the De Bruijn digraph with parameters `k,n`.
The De Bruijn digraph with parameters `k,n` is built upon a set of
vertices equal to the set of words of length `n` from a dictionary of
`k` letters.
In this digraph, there is an arc `w_1w_2` if `w_2` can be obtained from
`w_1` by removing the leftmost letter and adding a new letter at its
right end. For more information, see the
:wikipedia:`Wikipedia article on De Bruijn graph <De_Bruijn_graph>`.
INPUT:
- ``k`` -- Two possibilities for this parameter :
- An integer equal to the cardinality of the alphabet to use, that
is the degree of the digraph to be produced.
- An iterable object to be used as the set of letters. The degree
of the resulting digraph is the cardinality of the set of
letters.
- ``n`` -- An integer equal to the length of words in the De Bruijn
digraph when ``vertices == 'strings'``, and also to the diameter of
the digraph.
- ``vertices`` -- 'strings' (default) or 'integers', specifying whether
the vertices are words build upon an alphabet or integers.
EXAMPLES::
sage: db=digraphs.DeBruijn(2,2); db
De Bruijn digraph (k=2, n=2): Looped digraph on 4 vertices
sage: db.order()
4
sage: db.size()
8
TESTS::
sage: digraphs.DeBruijn(5,0)
De Bruijn digraph (k=5, n=0): Looped multi-digraph on 1 vertex
sage: digraphs.DeBruijn(0,0)
De Bruijn digraph (k=0, n=0): Looped multi-digraph on 0 vertices
"""
from sage.combinat.words.words import Words
from sage.rings.integer import Integer
W = Words(range(k) if isinstance(k, Integer) else k, n)
A = Words(range(k) if isinstance(k, Integer) else k, 1)
g = DiGraph(loops=True)
if vertices == 'strings':
if n == 0 :
g.allow_multiple_edges(True)
v = W[0]
for a in A:
g.add_edge(v.string_rep(), v.string_rep(), a.string_rep())
else:
for w in W:
ww = w[1:]
for a in A:
g.add_edge(w.string_rep(), (ww*a).string_rep(), a.string_rep())
else:
d = W.size_of_alphabet()
g = digraphs.GeneralizedDeBruijn(d**n, d)
g.name( "De Bruijn digraph (k=%s, n=%s)"%(k,n) )
return g
